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I am currently studying the textbook Physics of Photonic Devices, second edition, by Shun Lien Chuang. Section 2.1.1 Maxwell's Equations in MKS Units says the following:

The well-known Maxwell's equations in MKS (meter, kilogram, and second) units are written as $$\nabla \times \mathbf{E} = - \dfrac{\partial}{\partial{t}}\mathbf{B} \ \ \ \ \text{Faraday's law} \tag{2.1.1}$$ $$\nabla \times \mathbf{H} = \mathbf{J} + \dfrac{\partial{\mathbf{D}}}{\partial{t}} \ \ \ \ \text{Ampére's law} \tag{2.1.2}$$ $$\nabla \cdot \mathbf{D} = \rho \ \ \ \ \text{Gauss's law} \tag{2.1.3}$$ $$\nabla \cdot \mathbf{B} = 0 \ \ \ \ \text{Gauss's law} \tag{2.1.4}$$ where $\mathbf{E}$ is the electric field (V/m), $\mathbf{H}$ is the magnetic field (A/m), $\mathbf{D}$ is the electric displacement flux density (C/m$^2$), and $\mathbf{B}$ is the magnetic flux density (Vs/m$^2$ or Webers/m$^2$). The two source terms, the charge density $\rho$(C/m$^3$) and the current density $\mathbf{J}$(A/m$^2$), are related by the continuity equation $$\nabla \cdot \mathbf{J} + \dfrac{\partial}{\partial{t}}\rho = 0 \tag{2.1.5}$$ where no net generation or recombination of electrons is assumed.

I'm curious about this part:

where no net generation or recombination of electrons is assumed.

What does this mean in simpler terms? Why is this assumption necessary for $\nabla \cdot \mathbf{J} + \dfrac{\partial}{\partial{t}}\rho = 0$?

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The number/concentration of electrons in a volume may be due to their flow into / out of the volume (electric current), or due to the electrons appearing/disappearing inside of it. In vacumm, the latter possibility can be usually safely ignored (although not in QFT), so we have the continuity equation: $$\nabla\cdot\mathbf{J}+\partial_t\rho=0\Leftrightarrow \int_S\mathbf{J}\cdot\mathbf{ds} + \partial Q=0,$$ where the second equation is just the integral form of the continuity equation: the total current flowing through the surface surrounding the volume is the change of the charge within.

If, however, the charge may appear/vanish within the volume – which is a real option in semiconductors' interaction with the electromagnetic field – then we need to augment the continuity equation with a source term: $$\nabla\cdot\mathbf{J}+\partial_t\rho=s(t)$$

It is necessary to point out that the total charge conservation still holds (creation of an electron is accompanied by creation of a hole), but we would often want to describe electrons and holes separately – writing a continuity equation for each of them, or one type of the carriers may be quickly removed, and considered non-existent for the purposes of description (e.g., holes may be localized, but electrons highly mobile).

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  • $\begingroup$ How does your last equation relate to amperes law? (What would amperes law be if the continuity equation is this) And why doesnt this violate conservation of charge? Can I have some links to this derivation $\endgroup$ Commented Dec 7, 2021 at 12:28
  • $\begingroup$ @jensenpaull it does violate the conservation of charge - since the continuity equation does not hold. But, as I explained, we are not talking about all the charge. $\endgroup$
    – Roger V.
    Commented Dec 7, 2021 at 12:31
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    $\begingroup$ Ah yes, didn't even read the last paragraph before I commented as it seemed so obserd to me $\endgroup$ Commented Dec 7, 2021 at 12:34
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    $\begingroup$ @jensenpaull I added it later, after the first version of the answer was already published. $\endgroup$
    – Roger V.
    Commented Dec 7, 2021 at 12:35
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    $\begingroup$ @ThePointer It is a surface integrals. Whether to write it double is a matter of notation - double and triple quickly become impractical when working with functions of more variables. $\endgroup$
    – Roger V.
    Commented Dec 8, 2021 at 8:48

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